Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France.
Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia.
Phys Rev E. 2018 Aug;98(2-1):022219. doi: 10.1103/PhysRevE.98.022219.
The data recorded in optical fiber and in hydrodynamic experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrödinger equation (1D-NLSE) that governs at leading order the propagation of the optical and hydrodynamic waves in the two experiments. Nonlinear spectral analysis provides certain spectral portraits of the analyzed structures that are composed of bands lying in the complex plane. The spectral portraits can be interpreted within the framework of the so-called finite gap theory (or periodic inverse scattering transform). In particular, the number N of bands composing the nonlinear spectrum determines the genus g=N-1 of the solution that can be viewed as a measure of complexity of the space-time evolution of the considered solution. Within this setting the ideal, rational Peregrine soliton represents a special, degenerate genus 2 solution. While the fitting procedures previously employed show that the experimentally observed structures are quite well approximated by the Peregrine solitons, nonlinear spectral analysis of the breathers observed both in the optical fiber and in the water tank experiments reveals that they exhibit spectral portraits associated with more general, genus 4 finite-gap NLSE solutions. Moreover, the nonlinear spectral analysis shows that the nonlinear spectrum of the breathers observed in the experiments slowly changes with the propagation distance, thus confirming the influence of unavoidable perturbative higher-order effects or dissipation in the experiments.
利用非线性谱分析，对光纤和水动力实验中记录的数据进行了检查，这些数据报道了先驱性的观测，即具有时空局域化的非线性波类似于 Peregrine 孤子。我们的方法基于对一维聚焦非线性薛定谔方程（1D-NLSE）的可积性，该方程在两个实验中对光和水动力波的传播起主导作用。非线性谱分析为分析结构提供了某些谱像，这些谱像由位于复平面中的带组成。谱像可以在所谓的有限间隙理论（或周期逆散射变换）的框架内进行解释。特别是，组成非线性谱的带数 N 确定了可以看作是所考虑的解的时空演化复杂性的解的亏格 g=N-1。在这种设置下，理想的有理 Peregrine 孤子代表一种特殊的、退化的亏格 2 解。虽然之前使用的拟合程序表明，实验观察到的结构可以很好地近似为 Peregrine 孤子，但对光纤和水箱实验中观察到的呼吸子的非线性谱分析表明，它们表现出与更一般的、亏格 4 有限间隙 NLSE 解相关的谱像。此外，非线性谱分析表明，实验中观察到的呼吸子的非线性谱随传播距离缓慢变化，从而证实了实验中不可避免的微扰高阶效应或耗散的影响。
